Quadratic forms associated to stratifying systems☆

Let R be an algebra, and let (θ,≼) be a stratifying system of R-modules. If the category F(θ) is θ-directing, then we prove that indF(θ) is finite. In order to do that, we introduce a quadratic form qθ which depends on θ. Moreover, we also give sufficient conditions to get the correspondence X↦dim̲θX from indF(θ) to the set of positive roots of qθ.